login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A227135 Partitions with parts repeated at most twice and repetition only allowed if first part has an even index (first index = 1). 3
1, 1, 1, 2, 2, 4, 4, 6, 8, 10, 12, 17, 20, 25, 31, 39, 47, 58, 69, 85, 102, 123, 145, 175, 207, 246, 290, 343, 401, 473, 551, 646, 751, 875, 1012, 1177, 1358, 1570, 1807, 2083, 2389, 2746, 3140, 3597, 4106, 4690, 5337, 6082, 6907, 7848, 8895, 10085, 11404, 12902, 14561, 16438, 18520, 20864, 23460, 26385, 29619 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

FORMULA

Conjecture: A227134(n) + A227135(n) = A182372(n) for n>=0, see comment in A182372.

G.f.: 1/(1-x) + Sum_{n>=2} x^(A002620(n+2)-1) / Product_{k=1..n} (1-x^k), where A002620(n) = floor(n/2)*ceiling(n/2) forms the quarter-squares. - Paul D. Hanna, Jul 06 2013

EXAMPLE

G.f.: 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 8*x^8 +...

G.f.: 1/(1-x) + x^3/((1-x)*(1-x^2)) + x^5/((1-x)*(1-x^2)*(1-x^3)) + x^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + x^11/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) + x^15/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6))) +...

There are a(13)=25 such partitions, displayed here as partitions into two sorts of parts (format P:S for sort:part) where the first sort is 1 and sorts oscillate:

01:  [ 1:1  2:0  2:1  3:0  5:1  ]

02:  [ 1:1  2:0  2:1  4:0  4:1  ]

03:  [ 1:1  2:0  2:1  8:0  ]

04:  [ 1:1  2:0  3:1  7:0  ]

05:  [ 1:1  2:0  4:1  6:0  ]

06:  [ 1:1  2:0 10:1  ]

07:  [ 1:1  3:0  3:1  6:0  ]

08:  [ 1:1  3:0  4:1  5:0  ]

09:  [ 1:1  3:0  9:1  ]

10:  [ 1:1  4:0  8:1  ]

11:  [ 1:1  5:0  7:1  ]

12:  [ 1:1  6:0  6:1  ]

13:  [ 1:1 12:0  ]

14:  [ 2:1  3:0  3:1  5:0  ]

15:  [ 2:1  3:0  8:1  ]

16:  [ 2:1  4:0  7:1  ]

17:  [ 2:1  5:0  6:1  ]

18:  [ 2:1 11:0  ]

19:  [ 3:1  4:0  6:1  ]

20:  [ 3:1  5:0  5:1  ]

21:  [ 3:1 10:0  ]

22:  [ 4:1  9:0  ]

23:  [ 5:1  8:0  ]

24:  [ 6:1  7:0  ]

25:  [13:1  ]

MAPLE

## See A227134

# second Maple program:

b:= proc(n, i, t) option remember; `if`(n=0, 1-t,

      `if`(i*(i+1)<n, 0, add(b(n-i*j, i-1,

      irem(t+j, 2)), j=0..min(t+1, n/i))))

    end:

a:= n-> add(b(n$2, t), t=0..1):

seq(a(n), n=0..60);  # Alois P. Heinz, Feb 15 2017

PROG

(PARI) {A002620(n)=floor(n/2)*ceil(n/2)}

{a(n)=polcoeff(1/(1-x+x*O(x^n)) + sum(m=2, sqrtint(4*n), x^(A002620(m+2)-1)/prod(k=1, m, 1-x^k+x*O(x^n))), n)}

for(n=0, 60, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 06 2013

CROSSREFS

Cf. A227134 (parts may repeat after odd index).

Sequence in context: A057601 A294150 A087135 * A162417 A240012 A295261

Adjacent sequences:  A227132 A227133 A227134 * A227136 A227137 A227138

KEYWORD

nonn

AUTHOR

Joerg Arndt, Jul 02 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 22 13:09 EST 2018. Contains 299454 sequences. (Running on oeis4.)